Renormalization Group Approach To A Class Of Singular Differential Equations Disturbed By Stochastic Force

In 1990s,inspired by the quantum theory,Chen,Goldenfeld and Oono de-veloped a perturbative renormalization group(RG)method as a unified tool for asymptotic analysis.The basic idea of RG method is to obtain uniform-ly valid asymptotic solution of original problem by the one of RG equation-s.Compared with conventional methods(asymptotic match method,average method,multi-scale method,etc),RG method not only requires too many restrictions and assumptions,is but also useful for solving the approximate so-lution of variety of singular perturbation systems universally.However,there are few investigations for stochastic singular perturbed problems by utilizing RG method.The main work of this paper is to investigate a class of nonlinear stochastic singular perturbed problems.We obtain uniformly valid asymptotic solution of stochastic singular perturbed problem by constructing corresponding stochas-tic RG equation,and present the estimation of the error between the mild solution and the approximate one theoretically.Concretely,we consider initial value problem for stochastic differential equations as following:where ε is small parameter,Xε is ndimensional column vector A is diagonal matrix whose eigenvalues are either all nonnegative or all pure imaginary,F is n dimensional polynomial function column vector,G is n order polynomial function matrix,W is a standard n dimensional Brownian motion.There are 3 chapters in this paper.Chapter 1 is introduction,in which classical singular perturbation methods,RG method,stochastic singular per-turbation problems and main work are introduced.Chapter 2 is preparation work,where frequently-used symbols,an example towards RG method,funda-mental theory of stochastie differential equations and important lemmas are presented.Chapter 3 is main work,whose content is that we use RG method to obtain asymptotic solution whose uniform validity is proven later.